I Some confusion of Faraday's law of induction.

I have several confusions regarding Faraday's law of induction.
[tex]EMF = \int_{}^{} {\vec E \cdot d\vec l} = - \frac{{d\Phi }}{{dt}} = - \frac{d}{{dt}}\int_{}^{} {\vec B \cdot d\vec S} .[/tex] It means that If the magnetic flux Φ through the closed conducting loop changes in time, electric field E is induced along the loop, so the current flows.

Questions

1. Regarding the Faraday's law of induction, Lenz's law explains the induced current direction. It says the direction of induced current is such that the induced magnetic field B' from this current tends to oppose the change of Φ. This statement confused me to think that "secondary B-field", B', may also be included in the calculation of Φ when the Faraday's law is used. But I think magnetic field in the Φ is the field external to the loop, not including field generated from induction process, so B' must be excluded. Am I right?

2. Is Maxwell-Faraday equation [tex]\nabla \times \vec E = - \frac{{\partial \vec B}}{{\partial t}}[/tex] a special case of the Faraday's law? Stokes' theorem applying to the Maxwell-Faraday equation leads [tex]\int_{}^{} {\vec E \cdot d\vec l} = - \int\limits_{} {\frac{{\partial \vec B}}{{\partial t}}} \cdot d\vec S[/tex], which is not exactly same to the Faraday's law as shown above. It looks that It is a special case that magnetic field is time-varying while the loop is stationary (In fact, in the Maxwell-Faraday equation, a real loop is not even necessary, time-varying B-field just generates E-field no matter whether there is a real loop or not.)

3. Let's have a circular loop consisted of multiple elements, for example, metal wire + resistor, and the time-varying B-field is spatially uniform. In this case, is induced E-field uniform along the loop? I'm asking if there is any variation of induced E-field if the loop is made with segments of different materials.

2. They are equivalent: there are differential forms and integral forms of the equations
3. E is the same: R doesn't appear in the equations.

Well..in 2nd answer, how to derive the original Faraday equation from the differential form of MF (Maxwell-Faraday) Equation? Applying stokes' theorem to this MF equation only gives MF equation in an integral form where ∂/∂t is inside the integral, while the Faraday equation has d/dt outside the integral. In my view, the Faraday equation can handle not only time-varying B-field but also time-varying loop, while MF equation can only deal with time-varying B-field. Could you provide me a clear mathematical proof that two equations are truly equivalent?

In 3rd answer, yes, you're right. The line-integral doesn't care about the composition of the loop. So, we say that E-field is equivalent at all points on the loop. However, if this is true, a current density J = σE can be different at a different point as the conductivity σ varies. If the loop has an equal cross section all the way, the consequence is that a current is different at a different point!

Hmm...different currents lead accumulation of charges at some points so electrostatic field from the resulted spatial charge balances current?

I think this is valid only for a stationary boundary of integration (in a given inertial reference frame, of course), i.e. a stationary loop of wire. If the loop is moving, there is an extra term on the left which turns out to be the ##\vec v \times \vec B## part of the definition of ##\mathcal{E}## that I gave above. This is from my possibly imperfect memory, so I will be happy if someone corrects me if necessary. (I'm about to go out for the day so I don't have time to research further for now.)

Just to strengthen jtbell's posting, and this is very important and unfortunately not done right in most introductory textbooks that start with a very superficial treatment of the integral laws. One should keep in mind that from a modern point of view the basic laws are the local differential forms of Maxwell's equations. A lot of confusion with Faraday's Law can be avoided when using the differential fundamental laws and derive the complete integral forms. For arbitrary time dependent surfaces and boundearies this leads uniquely to the only correct integral law (written in Heaviside-Lorentz units):
$$\frac{1}{c} \frac{\mathrm{d}}{\mathrm{d} t} \int_{A} \mathrm{d}^2 \vec{A} \cdot \vec{B}=-\int_{\partial A} \mathrm{d} \vec{x} \cdot \left (\vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right),$$
where ##\vec{v}=\vec{v}(t,\vec{x})## is the velocity of the boundary ##\partial A##.

One should also not misunderstand the term "a time dependent magnetic field induces a electrical vortex field" in the sense of a causal connection. This idea is very hard to make sense of using the mathematics of the Maxwell equations. From a mathematical (and in my opinion also a physical point of view) the homogeneous Maxwell equations, i.e., Faraday's Law and Gauss's Law for the magnetic fied,
$$\vec{\nabla} \times \vec{E} + \frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0,$$
are constraints on the field components. Together with the inhomogeneous equations they lead to the explicit causal retarded solutions to the entire set of field equations, historically wrongly called the Jefimenko equations in the newer literature although they were be found much longer ago by Lorenz (without a t!) in the 1860ies. These equations clearly show that the causal sources of the electromagnetic fields are the electric charge-current distributions but not some components of the fields can be clearly interpreted as "causing" some other components.

I've summarized what I've studied for the relation between Faraday's law of induction (or simply called Faraday equation) and Maxwell-Faraday equation. I found that the Faraday equation describes not only Maxwell-Faraday equation, but also magnetic Lorentz force arisen by a movement of the closed loop in the magnetic field.

The attached file is the summary of my study so anybody sharing similar confusions would find it useful.